XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Stefan Rauch-Wojciechowski

Equations of motion of rolling and sliding axially symmetric rigid bodies and the motion of the Tippe Top

Abstract Equations of motion of a rigid body moving in the plane have simplest form written as Newton equations. We shall discuss the vector and coordinate form of these equations and their invariant manifolds. In case of pure rolling motion they admit 3 integrals of motion that are given by certain transcendental functions. They simplify in the case of rolling axially symmetric sphere modelling the tippe top.

The Tippe Top has a shape of a truncated sphere with a peg attached to the flat surface. When spun sufficiently fast on its spherical bottom the tippe top turns up and continues motion on the peg. This behaviour takes place for wide range of parameters and of initial conditions. We shall analyse the structure of the phase space through a sequence of invariant manifolds and give a description of what happens for all initial conditions and all values parameters. Then I show that, due to the gliding friction, all solutions tend (in the sense of the LaSalle´ theorem) to an asymptotic manifold consisting of periodic solutions. These solutions are Liapunov stable but are not asymptotically stable although every solution approaches one these periodic trajectories. I shall demonstrate the motion of the tippe top.